Abstract : A modular method for calculating hydrodynamic interactions between particles in low-Reynolds-number flow has been constructed by using multipole expansion solutions for the reflection field. The approach is made possible by the use of Faxen laws in relating the multipole moment to the incident field. The method is illustrated and checked by recalculating known expressions for the resistance and mobility tensors for two spheres. The method can be readily generalized to handle three-particle (or n-particle) interactions as shown in a following paper. New forms of the Faxen laws for prolate spheroids are given and will form the basis for other papers on spheroid-spheroid and spheroid-wall hydrodynamic interactions. The important result is that first-reflection solutions can be readily calculated even in cases where the ambient velocity field is obtained by a numerical procedure. These results, as asymptotic (far-field) solutions, furnish a check for more robust codes. In addition, these are important on their own since these provide crucial information for te renormalization theory used in suspension rheology.